(markov chain:)
for any two states A and Z, there is some n for which either the probability of reaching Z within n steps of starting from A is nonzero or the probabiltiy of reaching A within n steps of starting from Z is nonzero.

A reducible Markov chain is easily seen by its disconnected graph. For instance, if we have 2 states A and B, and P(A->A)=P(B->B)=1, then our graph is 2 disconnected circles, and obviously processes starting at A never read B. In this (extreme) case, any distribution is stable (it's not unique).

If, however, we chose P(A->B)=P(B->B)=1, our chain becomes irreducible, and it has a unique stable distribution: P(B)=1.

Def A polynomial over a field k (like the rational numbers, real numbers or complex numbers) is called irreducible if it is not a constant and cannot be factored as a product of polynomials (over k) of smaller degree.

More generally, an element of a commutative integral domain R is called irreducible if it is a non-unit and it cannot be written as a product of two non-units in R.

For example, in Z, the ring of integers, the irreducible elements are the prime numbers and their negatives.

If a is irreducible then any associate of a is irreducible. See also prime.

Now for a third mathematical definition of irreducible, this time from Representation Theory:
    Definition Suppose (ρ,V) is a representation of some group G. If there is no proper non-trivial subspace W of V such that ρ(G)W is contained in W, then the representation is said to be irreducible.
Come to think of it, Representation Theory is repsonsible for a great many overloaded definitions; simple is also an equivalent condition to irreducible, then there's complete, regular and characteristic which spring instantly to mind which also have definitions elsewhere in mathematics.
What is nice about the idea of irreducibility is that for finite and indeed compact groups, every representation of the group can be split up into a direct sum of irreducible ones, where the number of such irreducible representations is equal to the number of conjugacy classes of the group.

Ir`re*du"ci*ble (?), a.

1.

Incapable of being reduced, or brought into a different state; incapable of restoration to its proper or normal condition; as, an irreducible hernia.

2. Math.

Incapable of being reduced to a simpler form of expression; as, an irreducible formula.

Irreducible case Alg., a particular case in the solution of a cubic equation, in which the formula commonly employed contains an imaginary quantity, and therefore fails in its application.

-- Ir`re*du"ci*ble*ness, n. -- -- Ir`re*du"ci*bly, adv.

 

© Webster 1913.

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